Analysis of singularities of area minimizing currents

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In his monumental work in the 1980s, Almgren showed that for $n$-dimensional locally area minimizing rectifiable currents of codimension $> 1$, the singular set has Hausdorff dimension at most $n-2$. In contrast with codimension one, it is well-known that area minimizers of higher codimension can admit branch point singularities, at which at least one tangent cone is an $n$-dimensional plane with integer multiplicity $\geq 2$. Almgren’s lengthy proof consisted of first using an elementary argument based on tangent cone type to show that the set of non-branch point singularities has Hausdorff dimension at most $n-2$, and then using a powerful array of new ideas to bound the dimension of the set of all branch point singularities. In this strategy, the exceeding complexity of the argument stems largely from the lack of an estimate giving decay of $T$ towards a unique tangent plane at a branch point.

We will discuss a new approach (joint work with Neshan Wickramasekera) in which we first focus on showing uniqueness of tangent cones at $\mathcal{H}^{n-2}$-a.e. singular point, and then we study the size and fine structure of the singular set. As part of our method, we introduce an approximate monotonicity formula for a new intrinsic planar frequency function which quantifies the rate at which $T$ decays towards an $n$-dimensional plane. We decompose the singular set of $T$ into the set of branch points $\mathcal{B}$ at which $T$ decays to a unique tangent plane faster than a fixed rate (of radius to a power) and the set $\mathcal{S}$ of all remaining singular points, which a priori might contain a large set of branch points. Using the monotonicity of planar frequency functions together with the relatively elementary parts of Almgren’s proof (namely Lipschitz and harmonic approximation), we obtain locally uniform estimates indicating that near each point of $\mathcal{S}$ and at each sufficiently small scale, $T$ is significantly closer to some non-planar cone than it is to any plane. An analysis of singularities using the planar frequency function and locally uniform estimates recovers Almgren’s dimension bound for the singular set of $T$ in a simpler way. Using a blow-up methods of L. Simon and Wickramaskera and the locally uniform estimates for $\mathcal{S}$, we show that $T$ has a unique non-planar tangent cone at $\mathcal{H}^{n-2}$-a.e. point of $\mathcal{S}$, and thus $T$ has a unique tangent cone at $\mathcal{H}^{n-2}$-a.e. singular point. Again using the blow-up method together with the approximate monotonicity of the planar frequency function, we show that the entire singular set, including $\mathcal{B}$, is countably $(n-2)$-rectifiable and that $T$ has a unique homogeneous harmonic blow-up relative to its tangent plane at each point of $\mathcal{B}$.